Malmsten's Proof of the Integral Theorem

that Cauchy didn’t mention this property either in his treatment of definiteintegrals between real limits.Now Malmsten calls∫ zF (z) = f(z)dz (13)z 0and shows that F (z) is synectic in the same domain as f(z). This is analogousto what Cauchy showed for definite integrals between real limits,namely that F (z) is differentiable in the same domain as f(z).The last property to show isHere Malmsten uses expression (13) and putsF ′ (z) = f(z). (14)F (z + δ) − F (z) =∫ z+δzf(z)dzand then uses an earlier corollary (about weighted mean values) to getF (z + δ) − F (z) = δf(z + α) · θ · e pi ,where α is a ’mean quantity’ to 0 and δ (and therefore converges to 0 at thesame time as δ) and θ · e pi converges to 1 at the same time as δ convergesto 0. From this formula he getsF (z + δ) − f(z)lim = F ′ (z) = f(z).δHence, by proving the existence of the integral (1), and then showingthat the properties (10)-(14) holds, Malmsten has now completed his proofof the integral theorem (for integrals where the limits of integration arecomplex numbers) analogous to how Cauchy proved the integral theoremfor integrals between real limits.6 Final RemarksMalmsten left Uppsala at the end of the 1850s to get into politics (seeabove). His successor Herman Daug (1828-1888), whose mathematical researchmostly concerned differential geometry, didn’t have much influence[3]. Instead, in the 1870s, the mathematical education and research in Uppsalacame to be characterized by Göran Dillner (1832-1906), who had agreat interest in Cauchy’s works of analytic functions. But Dillner’s lacking20